The theory of the interaction between the conduction electrons and the quantum lattice vibrations (phonons) is a fundamental subject of research in solid state physics. The common description of the electron-phonon (el-ph) interaction in wide-band metals rests on the adiabatic approximation, in which the characteristic phonon frequency ω0 is orders of magnitude smaller than the Fermi energy EF of electrons. The adiabatic approximation allows to neglect all nonadiabatic vertex corrections of order λω0/EF or higher (where λ is the el-ph coupling), leading to the classical Migdal-Eliashberg theory of el-ph coupled systems.

In narrow-band solids, such as the fullerene compounds or magnesium diboride, the Fermi energy is of the same order of magnitude of the phonon frequency. In these systems, the adiabatic approximation is no longer valid and the classical Migdal-Eliashberg formalism is ill-defined, at least in principle. There is therefore the need to generalize the classical theory of the electron-phohon interaction in metals by taking inot account the non-adiabatic processes.

The main outcome of the non-adiabatic regime is the opening of extra channels in the electron-phonon interaction described by vertex-like corrections. Depending on material properties (charge carrier density, electron correlation etc.) these non-adiabatic channels may lead to an enhancement of the pairing interaction increasing the superconducting critical temperature Tc with respect to that expected in the adiabatic regime ω0/EF<< 1.

Giant spin-orbit coupling

Spin transport of itinerant electrons is a popular subject of research in view of the possible applications in spin-based electronic devices. One of the main subjects of study concerns the problem of controlling the spin relaxation of the charge carriers in order to maximise the length / time over which spin information is transferred / stored. In many systems of interest, spin relaxation is controlled by the Rashba spin-orbit coupling, which arises from the lack of inversion symmetry due to confining potentials in low dimensional systems or to non-centrosymmetric lattice potentials in bulk materials.

In the last decade, several low dimensional heterostructures and three-dimensional materials have been discovered to display giant Rashba couplings, of the order of 100 meV or more. The associate Rashba energy in these materials can thus be larger than other relevant energy scales, such as the Fermi energy or the phonon frequency. In this case, the Rashba interaction can have profound consequences both on the normal-state transport and on superconductivity, and it can give rise to new topological phases of matter.